Question

Extend the definition of conditional independence to more than $2$ events.

Short answer

Conditional independence is independence in conditional probability.

Step-by-step solution

Given Information 

The definition of conditional independence to more than $2$ events.

Explanation

Conditional independence of two events $-E_1,E_2$, with condition $F$is defined by any of two equivalent conditions:

$P\left(E_1\mid E_{2}F\right)=P\left(E_1\mid F\right)\iff P\left(E_1{E}_2\mid F\right)=P\left(E_1\mid F\right)P\left(E_2\mid F\right)$

Conditional independence is independence in conditional probability.

Generalize the formula for independence of multiple events:

$n$events $E_1,E_2,\dots ,E_n$are conditionally independent given $F$if

for any $k$and any different $j_1,j_2,\dots ,j_k\in \{1,2,\dots ,n\}$

$P\left({\cap }_{i=1}^k{E}_{j_i}\mid F\right)=\prod _{i=1}^{k}P\left(E_{j_i}\mid F\right)$

Final Answer

Multiple events are conditionally independent if conditional probability of intersection of any subset of events is the product of the conditional probability of those events.